Brownian dynamics simulation of needle chains

Abstract

Polymers consisting of rigid segments connected by flexible joints ͑needle chains͒ constitute an important class of biopolymers. Using kinetic theory as a starting point, we first derive the generalized coordinate-space diffusion ͑Fokker-Planck͒ equation for the needle chain polymer model. Next, the equivalent generalized coordinate Itô stochastic differential equation is established. Nonlinear transformations of variables finally yield a stochastic differential equation for the needle chain spatial coordinates in the laboratory coordinate system where the coefficients are expressed in terms of the chain constraint conditions. This latter equation constitutes the basis for our needle chain Brownian dynamics ͑BD͒ algorithm. The used needle chain model includes needle translation-translation and rotation-rotation hydrodynamic interactions, a homogeneous solvent flow field, external forces, excluded volume effects, and bending and twisting stiffness between nearest neighbor segments. For this chain model we find that by proper generalization of the involved parameters the mathematical analysis of the polymer dynamics, in great detail, maps onto the analysis of the bead-rod-spring polymer chain model with constraints presented by Ö ttinger in Phys. Rev. E 50, 2696 ͑1994͒. Preliminary numerical simulation data show that for a three segment needle chain, with needle axial ratio equal to five, our new needle chain BD algorithm is, in general, more than about 10 3 times more efficient than the bead-spring polymer chain BD algorithm commonly used as an approximation for studies of such polymer chains. This efficiency ratio increases asymptotically proportional to approximately the fourth power of the needle axial ratio. In addition to this major gain in efficiency, the needle chain model for segmented polymers, in general, incorporates a more realistic hydrodynamic description of the individual segments and, in particular, the joints between the segments than the bead-rod-spring models.

Transport properties of non-spherical nanoparticles studied by Brownian dynamics: theory and numerical simulations

“…Use of Eq. (15) yields that the determinant of the generalized mass tensor equals jm ) j ¼ I 2 k ðma 2 =2Þ 2 A 2 ðA 2 À 1Þsin 2 h 1 sin 2 h 2 ½1 À ðcos 2 nÞ=A 2 ;…”

Brownian dynamics simulations of needle chain and nugget chain polymer models—rigid constraint conditions versus infinitely stiff springs

“…This is essential to ensure that the divergence of the constrained mobility is included in the numerical scheme. 9,25 The integration scheme presented in Eqs. ͑59͒ and ͑60͒ is identical to the one presented earlier, 9,12 with one important exception, namely the metric force term F* (m,s) .…”

“…Algorithms for Brownian dynamics ͑BD͒ simulations of the motion of nugget chains were first introduced by Nyland et al 9 and Mikkelsen et al, 10 who connected the segments using rigid constraint conditions and springlike potentials, respectively. A generalized algorithm was presented by Klaveness and Elgsaeter.…”

“…12 Compared to bead chain models, the nugget chain algorithms represent a significant improvement in accuracy and/or computational efficiency for modeling a wide range of properties for semiflexible macromolecules. 9,10,12,13 If semiflexible molecules are to be represented accurately with bead chain models, a very high bending stiffness must be introduced in most interbead links. This requires a very short time step in the simulation algorithm to ensure numerical stability.…”

“…The combined effect of this can reduce the efficiency of the simulation algorithm by up to a factor of 10 3 compared to an algorithm where the motion of the stiff segments are simulated directly. 9,13 Orientational correlations introduced by rigid constraint conditions appear in nugget chain models, and they should thus be included in the BD algorithms. Until now this has not been fully understood, and their influence on stationary and dynamical properties of the models have been unknown.…”